Landmark XIV-C The joint two-line extension of one selected binary pair: noncoincident characteristic configurations, transverse estimates, and the output wavefront gate The retained characteristic diagnostic and the exact J2 stop

This manuscript continues XIII-C's joint-extension problem on the unchanged polynomial carrier. It does not continue XIV-B's relative antifield complex.

We freeze the switched metric–scalar vertex and the complete displayed second-order coefficient, including its scalar-sourced slot once. The eleven-branch decomposition, 48 labelled matchings, partial-diagonal ambiguity class, and elementary parallel-line extensions are copied. Individual line derivative orders distinguish those primitive results from the remaining joint problem.

The first new calculation is performed at a retained, noncoincident, null-collinear configuration. The three light-cone defining functions have differentials of rank two along a smooth characteristic stratum of codimension four. Their oriented conormals admit an explicit positive relation, reproducing the failure of the sufficient product criterion throughout this stratum. A retained component of the actual incidence has nonzero coefficient and nonvanishing differential symbols there. This does not determine whether the complete tensor and branch sum cancels the configuration.

A separate support lemma shows that the frozen collision contacts cannot alter any neighborhood disjoint from the collision loci. They therefore cannot supply an unnamed characteristic prescription. No uniform cancellation of the complete selected pair, and no prescription with the required ambiguity and wavefront control, is constructed. The first unresolved gate remains J2. The subsequent transverse, gluing, and output obligations are recorded but not opened.

The complete coefficient of $\lambda \hslash$ remains Unresolved. No nonexistence theorem, admitted quantum coefficient, residue subtraction, or Ward defect is asserted.

Figure fig:split-XIV-C records one view of the same separation. Figure fig:chain-XIV-C records one view of the same separation. Figure fig:pair-XIV-C records one view of the same separation. Figure fig:gate-XIV-C records one view of the same separation. The drawings add no hypothesis.

Figure fig:scope-XIV-C records that separation. The drawing adds no hypothesis.

and first stop{ Continuation: XIII-C, not XIV-B.

Frozen pair:

The scalar-linear lift ${{\Phi }^{(2)}}_{\mathrm{D}}$ alone is not silently identified with this eleven-branch observable.

Copied: the incidence inventory, ${\mathfrak{C}}_{\mathrm{part}}$, primitive parallel-line extensions, tree datum, and inherited support control.

Proved here: the rank-two characteristic geometry, its positive conormal relation, a nonzero retained component diagnostic, and the inability of collision-supported changes to modify a disjoint characteristic neighborhood.

First unresolved gate: J2. Neither uniform cancellation of the complete pair nor an admissible prescription through the characteristic set is supplied.

Ending for the joint coefficient: Unresolved. J3–J5 are not reached. No ${R_1}^{(1)}$ is admitted and no differential is applied to it.}

The three endings apply to the named analytic construction:

1. Constructed: the joint distribution exists with the prescribed normalizations, the four successor obligations are discharged, and its pushed-forward coefficient is admitted.

2. Proved obstructed: a demonstrated incompatibility excludes every successful construction in a previously frozen class.

3. Unresolved: a required prescription, estimate, contact composition, or output theorem has not been established.

Failure of a sufficient product criterion is not the second ending. A constructed elementary extension is not a joint extension.

For the frozen full displayed coefficient and switched vertex:

1. the eleven-branch decomposition and all 48 labelled double cross-contraction matchings are inventoried;

2. ${\mathfrak{C}}_{\mathrm{tot}}$ is a subclass of the widened partial-diagonal ambiguity class ${\mathfrak{C}}_{\mathrm{part}}$;

3. elementary parallel-line distributions admit extensions with their computed scaling-degree bounds;

4. collision exclusion and the $Z_3$ sum-zero bounds do not establish the complete joint product;

5. distinct null-related vertices admit a nonzero wavefront cancellation in a retained incidence;

6. the full coefficient of $\lambda \hslash$ remains Unresolved.

These are copied results, not new extension theorems of XIV-C XIIIC.

The four successor obligations are retained verbatim in substance:

1. a prescription through the noncoincident characteristic configurations, or proved cancellation in the selected pair;

2. uniform transverse estimates on the three- and four-vertex strata and their intersections;

3. compatible partial-diagonal contacts;

4. the output wavefront theorem after proper pushforward.

The vertex and the tree datum are admitted. The selected tree composition is

Its properness and compact microcausal kernel are inherited. The literal normalization $R_1={\mathcal{V}}_g{\star }_H\mathcal{O}$ is incompatible with the frozen retarded convention. The quantum joint extension was not released XIIIC.

Properness controls an existing distributional pushforward. It does not define an undefined product.

The supplied XIV-A and XIV-B manuscripts discuss a relative classical master equation, unrestricted classical breaking, compact relative ghosts and their free-shell boundary, and evaluation identities on smaller bosonic configuration sets.

None is used to define a propagator, a graph product, a contact, or an extension in this paper. No Koszul–Tate resolution, antifield algebra, or quantum interpretation of those manuscripts is needed here.

Likewise, the auxiliary value $-8I$ and the evaluation identity on ${\mathcal{C}}_{W,\mathrm{bos}}$ do not normalize this quantum coefficient. They are separate classical statements.

The distribution product criterion is used on domains where its hypothesis holds. It is sufficient, not a converse Hormander,FR.

Point-extension results control distributions already defined away from the point with finite scaling degree. Extension along a submanifold also requires longitudinal and microlocal control BF,Dang. Those tools do not provide the missing joint hypotheses merely by being cited.

Fix

Fix the inherited smooth ramp ${\sigma }_{\epsilon }$, zero for $t\le 0$ and one for $t\ge \epsilon$, the compact scalar switch $g$, the observation $f$, and the frozen translation-invariant Hadamard matrix $H$.

The carrier remains

The action Green blocks are

The metric–scalar blocks vanish. The inverse coefficient is $-1$, not the differently normalized X-track coefficient $-4$.

We denote by $F_{\mu }$ the scalar time-ordered numerical line of mass $\mu$, with the fixed nonzero species fiber factors separated when convenient. This notation does not change the frozen Hadamard data.

The predecessor notation distinguishes

from the full displayed coefficient

The former supplies nine branches; the latter supplies eleven. The target's instruction to retain the scalar-sourced slot and the inherited 48-match inventory therefore fixes the pair $({\mathcal{V}}_g,\mathcal{O})$, not the nine-branch pair $({\mathcal{V}}_g,{{\Phi }^{(2)}}_{\mathrm{D}})$.

This is a naming correction made before the calculation, not a change of observable after a detector. Throughout this paper, ${{\Phi }^{(2)}}_{\mathrm{D}}$ keeps its predecessor meaning.

The interaction is the functional

It is not an Euler source operator.

The inherited filled slots remain

The density-weighted source ${\mathcal{J}}_{\mathrm{D}}$ and its layer coefficients are those already selected and admitted on the XII/XIII track. No new Einstein–Hilbert insertion is added.

The retarded convention is

The requested order is

Here $\lambda$ counts one additional vertex insertion. It is not used to reweight the already filled slots.

The formal two-cross-line expression is

It remains formal until the complete graph compositions are defined. The difference is kept grouped; no undefined mixed intermediate product is declared to exist.

The inherited ordered values are

These refer to full functional supports.

For an inventory graph let $y$ be the interaction vertex. Connected collision subsets $I$ containing $y$ define

The inherited class includes their relevant intersections and contracted loci, while observable-only subgraph normalizations are retained.

Before pushforward, an allowed contact change has the form

subject to XIII-C's conditions: inherited smooth external weights, retained spectator graph, even parity, ghost number zero, two residual field slots, compact graph closure, and unchanged tree and observable-subgraph normalizations. Only defined products and pushforwards belong to ${\mathfrak{C}}_{\mathrm{part}}$.

The spectator need not be smooth in tangential variables. The bound six is the frozen conservative bound; an established degree can impose a smaller locus-specific bound.

The class controls permitted differences of constructions. It does not by itself select a reference distribution across the noncoincident characteristic set.

To ensure that the subsequent diagnostic belongs to the retained graph support, choose once

and the locations

Choose the original observation with $f(x_0)\ne 0$ and the original switch equal to one near these locations, with compact interior support. All switches and observations are fixed now.

This is an admissible retained instance of the inherited pair. No cutoff is applied to its numerical triangle, and no coefficient is chosen after computing a value. The subsequent diagnostic concerns this fixed instance; no assertion that every observation retains the same configuration is needed.

There is no restriction quotient, antifield-dependent observable, spurion transformation, convolution prescription, new Einstein–Hilbert vertex, triangle-residue subtraction, or sharp-width limit. Changing the pair, propagators, or ambiguity support after J2 would define a new contract.

This section names the inherited graph data. It constructs no new extension.

Write

where

The density coefficients are

Put

The full lift (eq:O) is ${{\sum }_{j=1}}^{11}{B}_j$, with

The signs, affine slot identity, and inclusion of the scalar-sourced slot once are inherited from XIII-C XIIIC.

A displayed divergence is transferred to its incoming numerical line. Derivatives on unused fields are not transferred to an unrelated line.

In the following table a superscript on a field slot is its derivative order, and an edge superscript is the derivative order on that internal retarded line. The scalar kinetic terms give the displayed maxima; the mass terms have the corresponding lower orders.

For the gravitational source terms, $|\alpha |+|\beta |\le 2$, including their fixed layer coefficients. For a stress monomial $e_s=1$ in the kinetic case and $e_s=0$ in the mass case.

The interaction slots at $y$ have orders

Therefore each cross-line order is fixed by its endpoints:

In $B_7$ the scalar cross-line can have order three. In $B_8$ and $B_9$ the metric cross-line carries the selected source derivative. For $B_{10}$ and $B_{11}$, each of the two scalar cross-lines separately uses (eq:cross-orders).

These assignments name line orders, not a tensor truncation. The complete graph has maximum total derivative count six after inclusion of the interaction.

For $j=1,\dots ,9$, label the two observable metric slots $i=1,2$ and the two interaction scalar slots $a=1,2$. The matchings

join the interaction metric to the selected observable metric and scalar slot $a$ to the single observable scalar. There are four for each branch.

For $j=10,11$, label the three observable scalar slots $i=1,2,3$. Select an unordered pair $i<k$ and one of its two bijections to the two interaction scalar slots:

There are six for each branch. Thus

Coincident slot locations do not erase the slot labels. Two metric cross-lines are absent because the interaction has only one metric slot; metric–scalar lines vanish for the frozen product data.

For a collapsed connected subset $I$, its normal codimension is $4(|I|-1)$. The inherited superficial count is $2E_I+D_I$, with numerical internal-edge count $E_I$ and derivative count $D_I$.

These are copied power counts, not established scaling degrees of a presently undefined joint distribution.

The copied primitive results concern

Their degrees satisfy

The temporal kinetic primitive has degree eight in codimension four and contact order at most four. These distributions already admit the stated point extensions XIIIC.

Their extension is not an extension of a three- or four-vertex composition with singular spectators. In particular it does not settle the noncoincident configuration examined next.

Let ${\mathcal{D}}_{\Gamma }$ be the relevant union of collision loci in a graph's vertex space. Define ${\mathcal{U}}_{\Gamma }$ to be the subset of its complement where the full graph product satisfies the distribution product criterion.

Every line is pulled back to the complete graph variables. The criterion excludes a selection of line wavefront covectors, with zero permitted for unused factors, whose sum vanishes at all vertices. It is not merely a test of the two cross-lines against each other.

On ${\mathcal{U}}_{\Gamma }$ the graph product is defined by the inherited product calculus. This does not establish equality

The following calculation identifies the retained failure of that proposed equality. No triangle cutoff is its starting point.

For the $B_6$ incidence, the underlying scalar numerical factors are

Choose

At the fixed triple in J0,

With the inherited time-ordered orientation, the wavefront selections at the line endpoints are

The retarded forward-null singularity permits both conormal signs. The time-ordered signs are those copied in XIII-C. A simultaneous reversal of Fourier convention reverses all three selections and leaves the calculation unchanged.

In the common vertex order $(x,u,y)$ these covectors are

They are nonzero and

The original weight $f(x_0){\sigma }_{\epsilon }(u_0)g(y_0)$ is nonzero. None of the vertices coincides with another. This is a retained configuration of the frozen graph, not a point removed by collision exclusion.

This selection is the inherited characteristic cancellation. The new calculation below identifies its geometric rank and support consequences.

Define

restricted to the open slab where all three vertices are interior. This stratum excludes every pairwise collision.

${\mathcal{N}}^{\circ }$ is a smooth embedded submanifold of the twelve-dimensional three-vertex space, of dimension eight and codimension four.

Let

where the square uses $\eta$. On ${\mathcal{N}}^{\circ }$,

Proof. The parameterization by $(u,\alpha ,\beta ,\mathbf{n})$ is smooth. It is injective, with smooth inverse on its image:

Its differential is injective: the $u$ components determine $\delta u$, the temporal differences determine $\delta \alpha$ and $\delta \beta$, and then the spatial differences determine the two tangent components of $\delta \mathbf{n}$. The dimension is $4+1+1+2=8$.

Writing $p_{\mathbf{n}}=\eta ({\ell }_{\mathbf{n}},\cdot )$ gives

These equations give the relation in (eq:rank). The last two differentials are independent because their nonzero components involve distinct free endpoint variations. The relation makes the rank exactly two.

The three light-cone equations consequently do not meet transversally at this stratum. Their differential rank is two although the null-collinear stratum has codimension four. Neither number is the codimension eight of a three-vertex collision diagonal.

At every point of ${\mathcal{N}}^{\circ }$, the oriented nonzero conormals

satisfy $n_R+n_1+n_2=0$ and reproduce (eq:nus) with $p=p_{\mathbf{n}}$.

There is no tangent vector $v$ at that point for which $n_R(v),n_1(v),n_2(v)$ are all strictly positive.

Proof. The formulas in the proof of Proposition prop:geometry give the stated covectors and sum. If all three evaluations were positive, their sum evaluated on $v$ would be positive, contradicting the zero relation.

The last assertion only rules out this simultaneous strict sign separation of the three selected conormals. It is not a theorem excluding other boundary-value constructions, finite parts, or cancellation in the complete pair.

The actual $B_6$ graph contains differentiated lines. The selected temporal term has, up to fixed nonzero constants,

Its differential orders are $1,0,2$, exactly the assignments of J1.

Before summation over tensor components and branches, the temporal incidence (eq:differentiated) has a nonzero coefficient in the frozen pair. Its line differential symbols are nonzero at the selections (eq:nus).

Proof. For a tensor whose only component is ${\gamma }_{00}=h$, the displayed density formula gives

Thus $B_6$ contains the nonzero monomial

Transferring the divergence produces a differentiated retarded line with a nonzero coefficient.

The interaction contains the temporal kinetic monomial $-\frac{1}{4}g{\gamma }_{00}({\partial }_0\varphi {)}^2$. One metric cross-contraction is nonzero because ${\mathcal{P}}_{00,00}=1/2$, and one scalar cross-contraction joins the two displayed temporal scalar derivatives. The fixed species factors are nonzero.

At (eq:nus), $p_0=1$. The temporal retarded derivative has nonzero symbol, and the two scalar derivatives have nonzero product symbol. Their principal conormal singularities are therefore not removed by these differential operators. Finally, the original external weight is nonzero at the retained triple fixed in J0.

This proposition verifies that the diagnostic is not an unrelated scalar toy graph. It does not prove that the full $B_6$ tensor sum, or the sum of all eleven branches, has a nonzero residue. No such residue has been defined.

To discharge J2 by cancellation one would have to construct or control the complete grouped coefficient and prove that the dangerous configurations disappear throughout the retained noncoincident characteristic domain.

A single nonzero component does not disprove cancellation between components. A single vanishing component does not prove it. The rank relation above is uniform as geometric data on ${\mathcal{N}}^{\circ }$; it is not a uniform theorem about the summed numerical distribution.

The inherited existence of the $H$ contraction also does not define the $H_T$ contraction. The grouped difference (eq:formal) must remain grouped until a valid prescription or cancellation argument is supplied.

The following support statement is proved at J2. It is not a J4 construction or gluing theorem.

For the three-vertex graph at $(x_0,u_0,y_0)$, there is an open neighborhood $V$ whose closure misses every collision locus of that graph.

Proof. The three locations are pairwise distinct. Their pairwise Euclidean coordinate distances have a positive minimum. Choose sufficiently small vertex neighborhoods so that their closures remain pairwise disjoint. Their product is the required $V$ and misses every equality of two vertices, hence every connected collision diagonal.

Every defined contact change in ${\mathfrak{C}}_{\mathrm{part}}$ for this graph restricts to zero on $V$. Consequently any two graph constructions whose difference belongs to the frozen ${\mathfrak{C}}_{\mathrm{part}}$ have the same restriction to $V$.

Proof. Each term (eq:Cpart) has support in its specified collision locus. Differentiation does not enlarge support; nor does multiplication in a defined product. The finite sum is supported in the collision union. Its restriction to the disjoint neighborhood $V$ is zero.

The statement is graphwise before pushforward, where the ambiguity class was defined. A contact can yield a nonlocal looking pushed-forward coefficient through its spectator; that does not change its pre-pushforward collision support.

If two proposed prescriptions differ on $V$ by a nonzero distribution supported on ${\mathcal{N}}^{\circ }\cap V$, their difference is not a member of the frozen collision-contact class.

Proof. Every class member vanishes on $V$ by Proposition prop:rigidity; the proposed difference does not.

This is a necessary ambiguity check, not nonexistence. A canonical or otherwise uniquely fixed characteristic prescription could in principle have no such extra ambiguity. No theorem establishes that prescription here.

In particular, “choose the partial contacts” is not a definition of the graph on $V$. The contacts can change collision extensions only after the off-collision graph, including its characteristic configurations, is defined.

Removing a neighborhood of ${\mathcal{N}}^{\circ }$ defines a smaller problem. A limit restoring that neighborhood needs an actual convergence or renormalized-limit theorem, its transverse degree, and its wavefront estimate.

No such limit is introduced. No cutoff triangle is assigned a value, and no subtraction supported on the characteristic set is added to ${\mathfrak{C}}_{\mathrm{part}}$. The class remains the one fixed in J0.

For the retained instance of the pair frozen in J0:

1. the separate-line product criterion fails on the noncoincident characteristic configuration and its smooth null-collinear stratum;

2. that failure is not removed by collision exclusion, line differentiation in the retained temporal component, or collision-supported contact changes;

3. neither a uniform cancellation theorem for the complete pair nor a prescription through the characteristic set with the required ambiguity and wavefront control has been constructed;

4. no contradiction excluding every possible successful construction in the frozen class has been proved.

The first unresolved gate is therefore J2, and its ending is Unresolved.

Proof. The copied wavefront selections and the new Proposition prop:geometry and Corollary cor:positive give (i). Proposition prop:component and Proposition prop:rigidity give the precise statements in (ii).

The calculations prove a failure of the sufficient criterion for the retained separate-line incidence and a restriction on permissible contact differences. They do not construct the complete branch sum across this domain, prove its cancellation, or define a characteristic finite part. Those missing constructions are exactly (iii). None of the proved statements is a converse to the distribution product criterion or a universal incompatibility of all prescriptions, giving (iv).

The copied parallel-line degree eight is a normal degree at a two-vertex collision. It supplies neither a normal degree along ${\mathcal{N}}^{\circ }$ nor a uniform degree at the larger collision strata and their intersections.

A future J3 proof must first have a distribution from J2. In tubular coordinates $(z,r)$ at a relevant stratum, it must control normal rescaling $r\mapsto \rho r$ uniformly over compact sets of longitudinal variables $z$. The rescaled distributions must have controlled test seminorms and the needed microlocal bounds.

If pointwise slices $t_z$ are used, their existence as distributional restrictions must also be proved; they are not automatic. Bounds at one fixed $z$ do not give the required uniform theorem. The extension literature explicitly separates these hypotheses from point-extension power counting BF,Dang.

No transverse degree for an undefined characteristic composition is assigned here. J3 is not opened.

The primitive contacts copied in J1 remain available on their established domains. A joint construction would have to multiply them by their singular spectators, impose the inherited subgraph normalizations, and prove agreement on every overlap.

No overlap agreement follows from the finite bound $|\alpha |\le 6$. No contact is selected after seeing the characteristic diagnostic to force a desired value.

Strict full-support ordering prevents an interaction attachment from coinciding with an observable attachment. Thus permitted collision contacts must leave those ordered regions unchanged, as copied from XIII-C. They must also leave the tree coefficient unchanged because they start at ${\hslash }^2$ in the time-ordered product. These remain normalization requirements, not a completed gluing proof. J4 is not opened.

The inherited compact graph closure and properness remain support inputs for a support-preserving construction. They do not turn an unnamed composition into a distribution.

The output target is a compact degree-two kernel with

under the fixed flat cotangent identifications. This is a sufficient microcausal bound: a nonzero tuple lying entirely in one pointed closed causal cone cannot have zero covector sum.

Assigning balanced covectors separately to edges is not the output proof. One needs a defined joint extension, its full graph wavefront theorem, and its actual proper pushforward. Since J2 is unresolved, none is supplied. No admission assertion for ${R_1}^{(1)}$ is made. J5 is not opened.

The following operations are not performed:

1. no $s_{\mathrm{ext}}$, $s_{\Psi }$, or $s_{\mathrm{free}}$ is applied to ${R_1}^{(1)}$;

2. no Hadamard line is contracted against ${\mathcal{J}}_{\mathrm{D}}$;

3. no binary Ward defect is formed;

4. no classical breaking density is identified with a quantum anomaly of this coefficient;

5. no value $-8I$ or evaluation identity on ${\mathcal{C}}_{W,\mathrm{bos}}$ is used as a subtraction;

6. no triangle residue is identified or subtracted.

The selected vertex has its own admitted local density. Its appearance in this binary problem is not a quantum retarded product of the gravitational cubic source.

XIV-A's action identity and XIV-B's relative complex do not enter the graph construction. Conversely the present characteristic diagnostic does not evaluate their classical breaking or observable ledger.

XIII-A's non-ideal restriction theorem also remains separate: no quotient by ${\mathcal{N}}_W$ is attempted. The source-switched X-track map and convolution-regulated coordinate problem are not used. All widths and switches remain fixed.

On the pair and ambiguity class frozen in J0:

1. the inherited inventory and primitive results remain valid as copied inputs;

2. the retained noncoincident characteristic configuration lies on a smooth codimension-four stratum with rank-two light-cone differentials and an explicit positive conormal relation;

3. permitted collision contacts cannot modify a neighborhood of that configuration;

4. these facts do not supply either cancellation of the complete pair or a joint characteristic prescription;

5. none of XIII-C's four successor obligations is fully discharged in this release.

The complete coefficient of $\lambda \hslash$ remains Unresolved, with the first stop at J2. No admitted ${R_1}^{(1)}$ and no universal extension obstruction are asserted.

Proof. Input statements input:XIIC, input:XIIIB, and input:branches establish the copied scope. Proposition prop:geometry, Corollary cor:positive, and Proposition prop:component establish the new diagnostic. Proposition prop:rigidity establishes contact rigidity. Theorem thm:J2 identifies exactly what those statements do and do not decide. The unentered obligations are recorded in Section sec:stop.

The stop is now geometrically specified: it is not a total or partial collision, but a nontransversal intersection of light-cone singularities at separated vertices. Its codimension and conormal relation are explicit.

The ambiguity issue is also separated. The inherited collision-contact class cannot serve as an unspecified choice of extension at that separated configuration. A future characteristic prescription must state why it has no forbidden additional ambiguity, or openly change the contract before using it.

These are analytic diagnostics and necessary conditions. They are not the missing distribution.

A successor receives:

1. the fixed full pair and the observable naming correction;

2. the copied branches, match labels, and line orders;

3. the retained configuration and characteristic stratum;

4. the rank relation (eq:rank), conormal selections, and nonzero temporal component diagnostic;

5. the support-rigidity theorem for partial contacts;

6. the exact J2 stop and the unchanged four obligations.

To advance beyond J2, the successor must either prove cancellation for the complete selected pair on the retained noncoincident domain or construct a prescription there with the declared normalization, degree, ambiguity, and wavefront control. Only afterwards can J3–J5 be entered.

It receives no quantum binary coefficient, Ward defect, interacting differential, reduced algebra, physical cohomology class, or numerical loop amplitude.

A failure of the product test is not nonexistence. A primitive extension is not the joint coefficient. An unresolved characteristic configuration remains unresolved. None of these statements is an interacting BRST algebra.

scope. The analytic predecessor is XIII-C. XIV-A and XIV-B are recorded only and are not hypotheses of any product or extension theorem here.

correction. The full eleven-branch observable is denoted $\mathcal{O}$. The scalar-linear lift ${{\Phi }^{(2)}}_{\mathrm{D}}$ retains its original meaning. The scalar-sourced slot is included once, before any diagnostic calculation.

scope. The branch inventory, primitive extension statements, partial-contact class, tree datum, and support control are copied from the supplied XIII-C handoff. The characteristic rank calculation, conormal relation, explicit component check, and contact-support rigidity are proved here.

scope. No joint characteristic distribution or cancellation theorem for the complete pair is supplied. J3–J5 are not opened. The manuscript is an exact analytic stop report, not a claim of completed renormalization.

scope. No renormalized quantum output, quantum master equation, Ward defect, state, residue subtraction, interacting differential, or sharp-width limit is constructed.

. No successful pdflatex execution, proof-assistant check, independently executed symbolic computation, independent referee review, or journal acceptance is asserted. The new calculations are the displayed analytic ones.

XIIIC

N. Cheung, Landmark XIII-C: The joint two-line extension of one selected binary pair, on the unchanged carrier. Supplied manuscript XIII-C.tex. Primary analytic handoff: branches, matchings, partial contacts, primitive extensions, characteristic configuration, tree inputs, properness boundary, and unresolved quantum coefficient.

XIVA

N. Cheung, Landmark XIV-A: Classical master equation and observable ledger for one declared Einstein–scalar sheet. Supplied manuscript XIV-A.tex. Recorded only; no analytic product or subtraction is imported.

XIVB

N. Cheung, {Landmark XIV-B: The relative antifield complex on ${\mathfrak{F}}_W$, and its trivial ghost sector on the free shell}. Supplied manuscript XIV-B.tex. Recorded only; its relative complex is not an input.

BF

R. Brunetti and K. Fredenhagen, “Microlocal analysis and interacting quantum field theories: Renormalization on physical backgrounds,” Communications in Mathematical Physics 208 (2000), 623–661, arXiv:math-ph/9903028. Sections 5–6: point extensions, transverse scaling, and microlocal extension along submanifolds.

Dang

N. V. Dang, “The extension of distributions on manifolds, a microlocal approach,” Annales Henri Poincaré 17 (2016), 819–859, arXiv:1412.2808. Extension and wavefront control across embedded submanifolds.

FR

K. Fredenhagen and K. Rejzner, Perturbative algebraic quantum field theory, arXiv:1208.1428. Distribution product criterion, microcausal functionals, and the distinction between Hadamard and time-ordered products.

Hormander

L. Hörmander, The Analysis of Linear Partial Differential Operators I, second edition, Springer, 1990. Distribution products, differential wavefront bounds, restriction, and pushforward criteria.